In the present study, the thermal energy storage of a hot petal tube inside a shell-tube type Thermal Energy Storage (TES) unit was addressed. The shell is filled with the capric acid Phase Change Material (PCM) and absorbs the heat from a hot U-tube petal. The governing equations for the natural convection flow of molten PCM and phase change heat transfer were introduced by using the enthalpy-porosity approach. An automatic adaptive mesh scheme was used to track the melting interface. The accuracy and convergence of numerical computations were also controlled by a free step Backward Differentiation Formula. The modeling results were compared with previous experimental data. It was found that the present adaptive mesh approach can adequately the melting heat transfer, and an excellent agreement was found with available literature. The effect of geometrical designs of the petal tube was investigated on the melting response of the thermal energy storage unit. The phase change behavior was analyzed by using temperature distribution contours. The results showed that petal tubes could notably increase the melting rate in the TES unit compared to a typical circular tube. Besides, the more the petal numbers, the better the heat transfer. Using a petal tube could increase the charging power by 44% compared to a circular tube. The placement angle of the tubes is another important design factor which should be selected carefully. For instance, vertical placement of tubes could improve the charging power by 300% compared to a case with the tubes' horizontal placement.